They do NOT stack.
If you have Expertise, taking a Feat to allow you to add double your proficiency bonus does not let you quadruple your Proficiency Bonus.
Can you stack proficiency and expertise?
It means that when you get Expertise from different classes, the number of skills on which you apply Expertise stacks. But Expertise can be applied only once on the same skill. You can get Expertise more than once, but you can never double your proficiency bonus on a roll more than once, as explained in the rules.
Can you have double proficiency and expertise 5e?
Expertise is always double proficiency, but double proficiency can be granted without calling it expertise. "You gain expertise with that skill, which means your proficiency bonus is doubled for any ability check you make with it."
